Analysis — page 1

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

eˣ and its tangent lines. The exponential curve with tangent lines at several points; at each point the slope equals the height.

The curve that is its own slope

There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.

Terms that vanish, a total that does not. The first 24 terms of the harmonic series as bars, with the running total above them. The last bar is 0.042 tall and the total has reached 3.776.

A sum whose terms vanish and whose total does not

Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.

Partial sums of sin x. sin x with its Taylor partial sums of degree 1, 3, 5, 9 about zero. Each extra term buys agreement over a wider interval and none of them is right everywhere.

One point's worth of information

A Taylor series claims that everything a function does, everywhere, is encoded in its behaviour at a single point. That claim is extraordinary, it is often true, and the cases where it fails are the interesting ones.

Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

The slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

The same terms, in a different order, adding to whatever is asked

Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

$A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.$

The sum that fits in one square

Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

The target, multiplied by one harmonic at a time. Four panels, each showing the square wave multiplied by a single sine. The areas cancel exactly except against the harmonics the wave actually contains.

Where the coefficients come from

The recipe for a square wave has a four over pi in front and a one over three on the second term, and the essay that built a square wave from sines used them without saying where they came from. They come from multiplying by one harmonic and taking the area.

The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

The corners go first

Fourier was not decomposing waves for the pleasure of it. He was solving the flow of heat, and the whole apparatus exists because each harmonic fades at a rate set by the square of its frequency — which is why a sharp profile smooths instantly and why the flow cannot be run backwards.

x ↦ cos x: two starts, one destination. A map whose graph is nowhere steeper than a fixed factor under one, with staircases from two different starting points converging on the same crossing, and the distance to it falling under a geometric bound.

A map that shrinks everything

One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

A curved map of the plane, and the flat one that fits it at a point. The map (x² − y², 2xy) carrying a small square patch of grid. Beside it, the image of the same patch under the linear map given by the matrix of partial derivatives, drawn dashed on top of the curved image.

The flat map that fits closest

A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.

eˣ and its inverse, reflected in the diagonal. A curve, the line y = x, and the curve reflected in it — which is the graph of the inverse function. Tangents are drawn at matched pairs of points, and the two slopes at each pair multiply to one.

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

A limit that forgets to be continuous

Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.

The area that names the number

The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.

The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.